Necessary conditions for the compensation approach for a random walk in the quarter-plane

Yanting Chen*, Richard J. Boucherie, Jasper Goseling

*Corresponding author for this work

Research output: Contribution to journalArticleAcademicpeer-review

Abstract

We consider the invariant measure of homogeneous random walks in the quarter-plane. In particular, we consider measures that can be expressed as a countably infinite sum of geometric terms which individually satisfy the interior balance equations. We demonstrate that the compensation approach is the only method that may lead to such a type of invariant measure. In particular, we show that if a countably infinite sum of geometric terms is an invariant measure, then the geometric terms in an invariant measure must be the union of at most six pairwise-coupled sets of countably infinite cardinality each. We further show that for such invariant measure to be a countably infinite sum of geometric terms, the random walk cannot have transitions to the north, northeast or east. Finally, we show that for a countably infinite weighted sum of geometric terms to be an invariant measure at least one of the weights must be negative.

Original languageEnglish
Pages (from-to)257-277
Number of pages20
JournalQueueing systems
Volume94
Issue number3-4
Early online date8 Jul 2019
DOIs
Publication statusPublished - 1 Apr 2020

Keywords

  • UT-Hybrid-D
  • Compensation approach
  • Geometric term
  • Invariant measure
  • Pairwise-coupled
  • Quarter-plane
  • Random walk
  • Algebraic curve

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